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用支持向量机（SVM）建模轻质泡沫混凝土的强度Abbas M. Abd Suhad M. Abd伊拉克迪亚拉大学工程学院文章历史：2016年9月13日收到2016年11月11日收到修订形式 2016年11月11日接受2016年11月14日在线提供摘要在为特定材料选择时，混凝土的强度是一个主要标准应用。这种建筑材料在经过很长一段时间后才获得实力浇注。在结构设计中考虑的普通混凝土的特征强度是定义为已经老化28天的样品的抗压强度。快速和对混凝土强度的可靠预测在经济上和实践上都是可行的意义重大。因此，对混凝土强度的预测一直是一个活跃的领域研究和相当多的研究已经开展。在这项研究中，两个技术被用来提出能够预测压缩的模型强度与可接受的准确性，这些是革命性的支持向量机（SVM）和多变量非线性回归。支持向量机模型被提出并且被开发用于预测早期混凝土抗压强度。预测模型中使用的变量来自混合比例元素和7天抗压强度的知识。这些模型提供了良好的压缩强度估计，并且具有良好的相关性。本研究中使用的数据与非线性多变量回归相关。此外，SVM模型被证明是预测抗压强度的重要工具，具有最小均方误差和标准偏差的轻质泡沫混凝土。关键词 泡沫混凝土；支持向量机；预测；抗压强度2016作者。这是由Elsevier有限公司出版在CC上开放获取文章BY-NC-ND许可证（/licenses/by-nc-nd/4.0/）。1.介绍混凝土被认为是全球最重要的建筑材料，也是最常用的材料建筑物或土木工程结构。 目前建筑业表现出色对使用轻质泡沫混凝土（LFC）作为建筑材料的兴趣，因为它具有许多有利的特性作为重量更轻，制造容易，耐用性和成本效益1。泡沫混凝土是新一代的轻质混凝土，具有一些吸引人的特点，如多功能其流动性，自密实性和自流平性，低尺寸变化和超低密度。 除此之外材料可设计成具有可控低强度，优异的绝热性能和良好的承重性容量，并且如果需要可以很容易地重新挖掘。凭借其独特的性能，泡沫混凝土有可能用于建筑的各种应用行业。 例如，Jones和McCarthy 2的一项研究调查了泡沫混凝土用作结构材料的潜力材料。 由于泡沫混凝土具有优异的隔热性能并且重量轻，所以可以与其他材料相辅相成材料将用于更高强度的结构应用2。在文献回顾中很好地同意组成材料和混合比例影响泡沫混凝土的性能和行为3-6。 不同构成材料对抗压强度的影响在文献和以往的研究中都得到了认可7-10。 发泡混凝土的抗压强度受密度，水泥类型和含量，水灰比，表面活性剂类型和固化方式的影响10。混凝土强度预测方法有几种强度预测关系是针对普通水泥浆，砂浆和混凝土开发的1,10。在目前建筑速度较快的情况下，很大程度上需要更多的混凝土生产，同时要注意混凝土的质量与标准和规格的一致性。必须在质量控制下生产出良好的混凝土，并且必须符合这些规范。规格通常指定测试方法以及测试的年龄。所有标准规定的混凝土强度非常重要（从1到28天），因为力量的早期发展（力量的早期增长）非常重要。但是，虽然混凝土的早期强度很重要，但后期的强度也很重要，因为毕竟这是混凝土结构设计中作为建筑材料所依赖的这种特性。传统的28天标准测试已被发现可以提供总体质量（在质量控制过程中使用）和混凝土验收的总体指标，并且在过去多年中表现良好。而且，速度很快，并尽可能早地对28天强度试验的结果进行可靠的预测会令所有各方满意，而不是等待传统的28天结果11。已经提出了许多改进的预测技术，包括经验或计算建模，统计技术和人工智能方法。统计技术：一些研究工作集中在使用多变量回归模型来提高预测的准确性。 统计模型具有吸引力，一旦拟合，它们可以用来比其他建模技术更快地进行预测，并且相应地在软件中实现更简单。 将统计建模与人工智能技术进行比较时尤其如此。 统计分析还可以通过相关分析提供对影响28天抗压强度的关键因素的洞察。 对于这些原因统计分析被选为本研究强度预测的技术。2.实验工作轻质泡沫混凝土的制造由四种材料组成，即普通波特兰水泥，沙子，水和泡沫。符合I型波特兰水泥的普通波特兰水泥（OPC）符合英国标准（BS EN 197-1：2000）中规定的要求。使用不同尺寸（600 m m，1.18和2 mm）（600 m m）的细硅砂和水（普通自来水）生产轻质泡沫混凝土。泡沫是一种稳定的气泡，通过在泡沫发生器中混合发泡剂和水而产生。泡沫的目的是通过将预先形成的稳定泡沫结合到新鲜轻质泡沫混凝土中来控制轻质泡沫混凝土的密度。在这项研究中，发泡剂与水的体积比为1:30。所用的超塑化剂是GLENIUM52，符合ASTM标准规范（ASTM C494M-04）。高效减水剂有深棕色水溶液。根据轻质泡沫混凝土的目标密度，w / c和s / c（沙/水泥比）设计最佳混合比例。密度范围为1500,1750和1800公斤/立方米。在这项工作中，所有混合物的w / c比率范围分别为0.5,0.45,0.4,0.35和0.3，而s / c为1.0。本研究中使用的固化方法是密封固化（包裹固化）。生产泡沫混凝土，然后倒入立方体中。在第7天和第28天测试150组混凝土立方体的密度和抗压强度。3.方法论3.1 支持向量机（SVM）支持向量机（SVM）是一种强大的监督学习算法，用于分类或回归12。 支持向量机是一种有区别的分类器，也就是说，它们绘制数据集群之间的边界。 支持向量机是基于定义决策边界的决策平面的概念。 决策平面是将一组具有不同类成员关系的对象分开的决策平面。 支持向量机（SVM）主要是一种分类方法，它通过在多维空间中构造超平面来执行分类任务，从而将不同类别标签的情况分开。 SVM支持回归和分类任务，并且可以处理多个连续和分类变量13。 对于分类变量，虚拟变量的创建案例值为0或1.因此，由三个等级（A，B，C）组成的分类因变量由一组三个虚拟变量表示：A：1 0 0，B：0 1 0，C：0 0 1为了构建最优超平面，SVM采用迭代训练算法，该算法用于最小化误差函数。 根据误差函数的形式，SVM模型可以分为四个不同的组：. 分类支持向量机类型1（也称为C-SVM分类）. 分类支持向量机类型2（也称为nu-SVM分类）. 复原SVM类型1（也称为-SVM复原）. 复原SVM类型2（也称为nu-SVM复原）3.2复原SVMAbubakar等人 （2013）指出，在回归支持向量机中，因变量y对一组自变量x的函数依赖性必须进行估计。 与其他回归问题一样，它假定独立变量和因变量之间的关系由确定性函数f加上一些加性噪声14给出：y = f（x）+噪声然后，任务是为f找到一个函数形式，它可以正确预测SVM之前没有提交过的新案例。 这可以通过在样本集上训练SVM模型来实现，即训练集，涉及分类的过程（参见上文）以及错误函数的顺序优化13,15。 根据此误差函数的定义，可以识别两种类型的SVM模型：3.3 复原SVM类型1对于这种类型的SVM，错误函数是：12WTW+Ci=1Ni+Ci=1Ni错误功能被最小化，受限于：WTXI+b-yi+iyi-WTXI-bi+iii0,i=1,N支持向量机模型中可以使用多个内核。 这些包括线性，多项式，径向基函数（RBF）和S形：3.4 内核函数KXIXJ= XiXJ 线性 (XiXJ+C)d 多项式exp-XI-XJ2 RBFtanhXiXJ+C S状弯曲位置K XiXJ= Xi*(XJ)也就是说，核函数表示通过变换f映射到高维特征空间的输入数据点的点积。 Gamma是某些内核函数的可调参数。 RBF是迄今为止支持向量机中最常用的内核类型选择。 这主要是因为它们在实际x轴的整个范围内的局部和有限响应。3.5 径向基函数（RBF）它是一个实值函数，其值仅取决于与原点的距离，因此f（X）= f（XX）; 或者也可以在距离某个其他点c（称为中心）的距离上，以使得f（X，C）= f（XC）。 任何满足f（X）= f（X）性质的函数都是径向函数。 范数通常是欧几里得距离，尽管其他距离函数也是可能的。 例如，使用Lukaszyk-Karmowski度量，对于一些径向函数来说，有可能避免因解决矩阵问题而导致的问题，从而确定系数wi，因为X总是大于零16,17。径向基函数的和通常用于近似估计给定函数。 这个近似过程也可以被解释为一种简单的网络。 2014年，Preetham等人提出了与土木工程相关的支持向量机方法（SVM）问题。 显示了许多研究领域正在进行的关于SVM技术的数值研究。 RBFs的许多研究也被用作支持向量分类的核心16。4.结果和讨论4.1 性能轻质泡沫混凝土测试了这项工作中轻质泡沫混凝土样品的新鲜干密度，7天和28天抗压强度。 根据以前的工作和其他研究人员，考虑到混合比例的参数是影响泡沫混凝土抗压强度的参数7-10。 据信密度是影响泡沫混凝土抗压强度的关键因素，因为泡沫混合物中加入的泡沫量控制了混合物的密度，从而控制了其强度。 混合。 这归因于增加由发泡引起的气泡的事实，添加到混合物中的添加剂会增加孔隙率，同时削弱其强度。 图1说明了这一事实，其中轻质泡沫混凝土的抗压强度与其干密度之间的关系。另一方面，如图2所示，增加轻质泡沫混凝土的水泥含量可提高其抗压强度（特别是因为泡沫混凝土不包括粗骨料，只有细骨料）解释这一趋势是，通过增加水泥，与水反应的优质材料增加，导致更多的水合产物和对混合物的结合，这增加了强度。 另外，发现细骨料粒径的增加会降低其强度（表1）4.2 第一：传统的多变量非线性回归为了预测泡沫混凝土的28天抗压强度，使用非线性回归来分析150个样品的数据集。 在这项研究中，用于模拟28天抗压强度的主要变量是： 密度，水泥含量，砂含量，w / c比，砂粒大小，发泡剂，泡沫含量和7天时的抗压强度。 28天抗压强度非线性回归的一般模型为：Var10 = a0*v1a1*v2a2*v3a3*v4a4*v5a5*v6a6*v7a7*v8a8*v9a9Var10（因变量）= 28天时的抗压强度V1至V9（独立变量）=输入参数本分析中使用的损失函数是最小二乘法。当实际观察与使用开发模型的预测结果相比时，相关系数被发现R = 0.97884248和r2 = 0.9581326置信水平的限制是：95%（a= 0.050）。 表2列出了各模型参数（a n）与标准差，t值和p值的系数。实际观察结果与图3（a）中的回归模型生成的预测结果一起绘制。 该图解释了两个数据集之间的高度相关性，并反映了所开发模型的高精度。很少有实际观测中，围绕30MPa的抗压强度的点很少有分歧。 这可能属于原材料的性质（特别是沙粒的大小）和被测样品的具体情况。用预测结果绘制残差值反映了开发模型的良好性能。 再根据预测的抗压强度，该曲线图精确显示每次读数的误差量。 在30 MPa附近的区间非常明显，实际观测值有一定的收敛性，同时大部分结果误差在（-4和+4）之间，如图3（b）所示。 9次投入和目标产出的总体相关系数表明，与第28天抗压强度的最高相关性与第7天抗压强度相关。图1发泡混凝土的抗压强度与密度的关系图2发泡混凝土的抗压强度与水泥含量的关系表格1混合比例细节混合样品数量S / C比w / c比密度范围（Kg / m 3）SP（L）砂最大尺寸（毫米）沙子类型C11810.451400-200000.6硅砂C2180.50.451400-200000.6硅砂C31820.451400-200000.6硅砂C41810.41400-200000.6硅砂C51810.351400-20000.20.6硅砂C61810.31400-20000.30.6硅砂C71810.351400-20000.30.6河洗沙C81810.351400-20000.41.18河洗沙C9610.41400-20000.54.75河洗沙密度和水泥含量分别为正相关。 它与负相关分别与水灰比，砂/水泥比和泡沫含量有关。表2回归模型的系数（参数a）参数变量估计标准-错误t值- df = 140p值a 00.00004100.338570.735442a 1密度1.132920.581.96840.050997a 2水泥0.3370861506.590.000010.999996a 3沙0.3370861506.550.000010.999996a 4沙/水泥-0.179661506.5700.999998a 5水/水泥0.3284830.152.246870.026213a 6沙大小-0.214110.02-7.503640a 7媒介-0.076560.03-3.818340.000201a 8泡沫0.1075720.061.720060.087631a 9COMP-7D0.53750.0511.180104.3 第二：支持向量机为了实施这项技术，28天时的抗压强度被认为是因变量（Var10），而其他输入（V1到V9）被认为是自变量。 将150次整体观察的样本大小随机分成（111个样本）和（39个样本）的测试。 分析过程采用类型1的支持向量机。 测试了四种核函数类型：径向基函数，线性函数，多项式函数和S形函数。 这个过程结果列在表3中。显然，就训练，测试和整体数据集的相关性而言，RBF具有最好的结果。 它具有四个函数中的最小均方误差，并具有最小的标准偏差。 所以详细的讨论将集中在RBF来解释这个模型的主要特征。 表4说明了所有调查样本的预测模型中的总误差均值（-0.32084）。 整体相关系数非常显着（约99），这反映了所开发模型的高度精确度，如图4所示。(a) 抗压强度（预测值与观测值）(b) 抗压强度（预测结果与残余）图3 回归模型的性质表3支持向量机的四种类型函数结果功能类型相关系数均方误差标准偏差径向基函数0.986（训练），0.990（测试），0.987（总体）3.880（训练），3.268（测试），3.721（总体）0.170（训练），0.147（测试），0.165（总体）线性0.951（训练），0.945（测试），0.949（总体）18.444（训练），25.263（测试），20.217（总体）0.369（训练），0.413（测试），0.381（总体）多项式0.976（训练），0.986（测试），0.978（总体）6.714（训练），5.357（测试），6.361（总体）0.225（训练），0.178（测试），0.215（总体）S状弯曲0.851（训练），0.877（测试），0.859（总体）67.969（训练），66.761（测试），67.655（总体）0.716（训练），0.673（测试），0.703（总体）表4 RBF支持向量机模型的主要特征支持向量的数量30（16有界），（= 0.111）模型规范（决策常数）0.124238观测平均数26.90346预测平均数27.22430观测S.D12.28756预测S.D11.54473均方误差3.26776误差均值-0.32084误差S.D1.80225ABS.误差均值1.49713S.D比0.14667关联0.99（a） 训练数据集（b） 测试数据集（c） 整体数据集图4（a）训练数据，（b）测试数据和（c）总体数据集的相关图（a）测试数据组（b）和总体数据组（c）的RBF的预测值绘制观察数据组。 非常清楚的是，预测值的分布非常接近数据中所有数据集的平等线并且非常接近与实际观察到的数据相关，表明上述模型的可靠性很高。 对于每个数据集，最佳拟合的公式都提供了它的图。5 结论这项工作揭示的结果包括混合比例对轻质泡沫混凝土28天抗压强度的影响。 密度和水泥含量的正面影响非常明显，并且证明这两个因素在设计泡沫混凝土混合料中具有重要和重要的作用。 同时，发泡混凝土的抗压强度对混凝土的抗压强度有较大的负面影响，分别表现为w / c比，砂/水泥比和泡沫含量的增加。本研究提出了预测轻质泡沫混凝土抗压强度的数学模型。 用于执行提出的模型的技术是传统的多变量非线性回归和革命性的支持向量机建模。 结果显示在本研究中使用的数据集的观察值和预测值之间具有极好的相关性。 这两种技术都被证明是预测过程的有吸引力的工具。 采用径向基函数RBF的SVM技术与其他函数和传统回归方法相比，预测结果的最小均方误差和标准差较大。 这反映了该工具沿着总体相关数据集预测结果中所有点的高精度。参考文献1 Md Azree Othuman Mydin, Potential of using lightweight foamed concrete in composite load-bearing wall panels in low-rise construction, Concr. Res.Lett. 2 (2) (2011) 213227.2 M.R. Jones, A. McCarthy, Preliminary views on the potential of foamed concrete as a structural material, Mag. Concr. Res. 57 (1) (2005) 2131.3 M.S. Hamidah, I. Azmi, M.R.A. Ruslan, K. Kartini, N.M. Fadhil, Optimisation of foamed concrete mix of different sand-cement ratio and curingconditions, in: R.K. Dhir, M.D. Newlands, A. McCarthy (Eds.), Proc. of the International Conference on the Use of Foamed Concrete in Construction,Thomas, London, 2005, pp. 3744.4 M.R. Jones, A. McCarthy, Heat of hydration in foamed concrete: effect of mix constituents and plastic density, Cem. Concr. Res. 36 (6) (2006) 10321041.5 E.P. Kearsley, P.J. Wainwright, The effect of high fly ash content on the compressive strength of foamed concrete, Cem. Concr. Res. 31 (1) (2001) 105112.6 E.P. Kearsley, P.J. Wainwright, Ash content for optimum strength of foamed concrete, Cem. Concr. Res. 32 (2) (2002) 241246.7 E.K.K. Nambiar, K. Ramamurthy, Models relating mixture composition to the density and strength of foam concrete using response surfacemethodology, Cem. Concr. Comp. 28 (9) (2006) 752760.8 E.K.K. Nambiar, K. Ramamurthy, Models for strength prediction of foam concrete, Mater. Struct. 41 (2) (2008) 247254.9 K. Ramamurthy, E.K. Kunhanandan Nambiar, G. Indu Siva Ranjani, A classification of studies on properties of foam concrete, Cem. Concr. Compos. 31 (6)(2009) 388396.10 Suhad M. Abd, M.F.M. Zain, Roszilah Hamid, Abbas M. Abd, Fuzzy modelling system to predict the compressive strength of concrete, Proc. of NationalSeminar on Fuzzy Theory and Fuzzy: From Theory to Applications, UiTM University, Malaysia, 2008, pp. 116.11 G.F. Kheder, A.M. Al Gabban, S.M. Abid, Mathematical model for the prediction of cement compressive strength at the ages of 7 and 28 days within24 hours, Mater. Struct. 36 (December) (2003) 693701.12 C. Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery, Volume (2, Kluwer AcademicPublishers, Boston, 1998.13 O. Chapelle, V. Vapnik, O. Bousquet, S. Mukherjee, Choosing multiple parameters for support vector machines, Mach. Learn. 46 (2002) 131159.14 Abubakar S. Magaji, Audu Isah, Victor Onomza Waziri, K.R. Adeboye, A conceptual Nigeria stock exchange prediction: implementation using supportvector machines-SMO model, World Comput. Sci. Inf. Technol. J. (WCSIT) 3 (4) (2013) 8590.15 S. Lukaszyk, A new concept of probability metric and its applications in approximation of scattered data sets, Comput. Mech. 33 (2004) 299304.16 S. Preetham, M. Shivaraj, W.P. Prema Kumar, H. Ravi Kumar, Support vector machines technique in analysis of Concrete- critical review, Int. J. Emerg.Technol. Eng. (IJETE) 1 (9) (2014) 199203.17 Abbas M. Abd, Suhad M. Abd, Neuro-Fuzzy model to evaluate ready-Mix concrete properties, IJERST 3 (1) (2014) 2433.附录A.补充数据与本文相关的补充数据可以在网上找到，网址为/10.1016/j.cscm.2016.11.002。Case studyModelling the strength of lightweight foamed concrete usingsupport vector machine (SVM)Abbas M. Abd*, Suhad M. AbdCivil Engineering Department, College of Engineering, Diyala University, IraqA R T I C L E I N F OArticle history:Received 13 September 2016Received in revised form 11 November 2016Accepted 11 November 2016Available online 14 November 2016Keywords:Foamed concreteSupport vector machinePredictionCompressive strengthA B S T R A C TStrength of concrete is a primary criterion in selecting this material for a particularapplication. This construction material gains strength over a long period of time afterpouring. Characteristic strength of normal concrete that considered in structural design isdefined as the compressive strength of a sample that has been aged for 28 days. Rapid andreliable prediction for the strength of concrete would be economically and practically ofgreat significance. Therefore; the prediction of concrete strength has been an active area ofresearch and a considerable number of studies have been carried out. In this study, twotechniques were used to propose a model which is capable of predicting the compressivestrength with acceptable accuracy, these were the revolutionary support vector machine(SVM) and the multivariable non-linear regression.Support vector machine model was proposed and developed for the prediction ofconcrete compressive strength at early age. The variables used in the prediction modelswere from the knowledge of the mix proportion elements and 7-day compressive strength.The models provide good estimation of compressive strength and yielded good correlationswith the data used in this study relative to nonlinear multivariable regression. Moreover,the SVM model proved to be significant tool in prediction compressive strength oflightweight foamed concrete with minimal mean square errors and standard deviation. 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CCBY-NC-ND license (/licenses/by-nc-nd/4.0/).1. IntroductionConcrete is considered worldwide as the most important building material and also the most common material used inthe construction of buildings or civil engineering structures. Presently the construction industry has shown significantinterest in the use of lightweight foamed concrete (LFC) as a building material due to its many favourable characteristics suchas lighter weight, ease of fabrication, durability and cost effectiveness 1.Foamed concrete is a new generation of lightweight concrete that is versatile with some attractive characteristics such asits flowability, self-compacting and self-levelling nature, low dimensional change and ultra-low density. In addition, thematerial can be designed to have controlled low strength, excellent thermal insulation properties, and good load-bearingcapacity and can be easily re-excavated, if necessary.With its unique properties, foamed concrete has the potential to be used in various applications in the constructionindustry. For example, a study by Jones and McCarthy 2 investigated the potential of foamed concrete for use as a structural* Corresponding author.E-mail address: abbas.mahde (A.M. Abd)./10.1016/j.cscm.2016.11.0022214-5095/ 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (/licenses/by-nc-nd/4.0/).Case Studies in Construction Materials 6 (2017) 815Contents lists available at ScienceDirectCase Studies in Construction Materialsjournal homepa ge: /locate/cscmmaterial. Since foamed concrete has excellent thermal insulating properties and is lightweight, it can complement othermaterials to be used in higher strength structural applications 2.It is well agreed in the literature review that constituent materials and mix proportions affect the properties andbehaviour of foamed concrete 36. The possible effect of different constituent materials on the compressive strength hadbeen recognized in the literature and through past research 710. The compressive strength of foamed concrete is affectedby the density, cement type and content, water/cement ratio, surfactant type and curing regime 10.1.1. Prediction methods for strength of concreteThere are several strength prediction relations developed for plain cement paste, mortar and concrete 1,10. Under thecurrently quicker pace of construction, there was a great need for more production of concrete with attention to theconformability of the quality of the produced concrete with the standards and specifications. Good concrete must beproduced under quality control and must comply with these specifications. Specifications usually specify a test method aswell as age of test. Strength of concrete, as specified by all the standards, is very important (from 1 to 28 days), because theearly development of strength (early gain in strength) is very important. But, while early strength of concrete is important,strength at later ages is also important, because after all, it is this property which is relied upon in structural design ofconcrete as a construction material. The traditional 28 days standard test has been found to give a general index of the overallquality (used in quality control process) and acceptance of concrete and has served well for so many years. Moreover, rapidand reliable prediction of the results of 28 days strength test as early as possible would be of satisfaction for all parties insteadof waiting for the traditional 28 days results 11. A number of improved prediction techniques have been proposed byincluding empirical or computational modelling, statistical techniques and artificial intelligence approaches.Statistical techniques: A number of research efforts have concentrated on using multivariable regression models toimprove the accuracy of predictions. Statistical models have the attraction that once fitted they can be used to performpredictions much more quickly than other modelling techniques and are correspondingly simpler to implement in software.This is especially true when comparing statistical modelling with artificial intelligence techniques. Statistical analysis canalso provide insight into the key factors influencing 28 days compressive strength through correlation analysis. For thesereasons statistical analysis was chosen to be the technique for strength prediction of this study.2. Experimental workThe making of lightweight foamed concrete consist of four types of material, namely ordinary Portland cement, sand,water and foam. Ordinary Portland cement (OPC) complied with Type I Portland Cement complies with the requirementsspecified in the British Standard (BS EN 197-1: 2000). Fine silica sand of different sizes (600 mm, 1.18 and 2 mm) (600 mm)and water (normal tap water) was used in producing the lightweight foamed concrete. The foam is a form of stable bubbles,produced by mixing foaming agent and water in a foam generator. The purpose of the foam is to control the density oflightweight foamed concrete by incorporating preformed stable foam into fresh lightweight foamed concrete. For this study,the ratio of foaming agent to water is 1:30 by volume. The superplasticizer used was GLENIUM52, conforming to the ASTMstandard specification (ASTM C494M04). The superplasticizer is available in dark brown aqueous solution. The optimummix proportion was designed based on target density, w/c and s/c (sand to cement ratio) of lightweight foamed concrete. Therange of densities were 1500, 1750 and 1800 kg/m3. The range of w/c ratio used were 0.5, 0.45, 0.4, 0.35 and 0.3, while s/c was1.0 for all mixes in this work. The method of curing used in this study was sealed curing (wrapped curing). Foamed concretewas produced and then poured in cubes. 150 sets of concrete cubes were tested for their density and compressive strength at7 and 28 days.3. Methodology3.1. Support vector machines (SVM)Support Vector Machines (SVMs) are a powerful supervised learning algorithm used for classification or for regression12. SVMs are a discriminative classifier: that is, they draw a boundary between clusters of data. Support Vector Machinesare based on the concept of decision planes that define decision boundaries. A decision plane is one that separates between aset of objects having different class memberships. Support Vector Machine (SVM) is primarily a classier method thatperforms classification tasks by constructing hyperplanes in a multidimensional space that separates cases of different classlabels. SVM supports both regression and classification tasks and can handle multiple continuous and categorical variables13. For categorical variables a dummy variable is created with case values as either 0 or 1. Thus, a categorical dependentvariable consisting of three levels, say (A, B, C), is represented by a set of three dummy variables:A: 1 0 0, B: 0 1 0, C: 0 0 1To construct an optimal hyperplane, SVM employs an iterative training algorithm, which is used to minimize an errorfunction. According to the form of the error function, SVM models can be classified into four distinct groups:A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 815 9? Classification SVM Type 1 (also known as C-SVM classification)? Classification SVM Type 2 (also known as nu-SVM classification)? Regression SVM Type 1 (also known as epsilon-SVM regression)? Regression SVM Type 2 (also known as nu-SVM regression)3.2. Regression SVMAbubakar et al. (2013) stated that in a regression SVM, the functional dependence of the dependent variable y on a set ofindependent variables x has to be estimated. It assumes, like other regression problems, that the relationship between theindependent and dependent variables is given by a deterministic function f plus the addition of some additive noise 14:y = f(x) + noiseThe task is then to find a functional form for f that can correctly predict new cases that the SVM has not been presentedwith before. This can be achieved by training the SVM model on a sample set, i.e., training set, a process that involves, likeclassification (see above), and the sequential optimization of an error function 13,15. Depending on the definition of thiserror function, two types of SVM models can be recognized:3.3. Regression SVM type 1For this type of SVM the error function is:12wTw CXNi1ji CXNi1j?iThe error function was minimized subject to:wTfxi b ? yi? e ji?yi? wTfxi ? bi? e jiji; ji? 0; i 1; :; NThere are several numbers of kernels that can be used in Support Vector Machines models. These include linear,polynomial, radial basis function (RBF) and sigmoid:3.4. Kernel functionsK Xi; Xj? ?Xi: XjLineargXi:Xj C? ?dPolynomialexp ?gjXi?Xjj2? ?RBFtanh gXi:XjC? ?Sigmoid8:9=;Where K Xi; Xj? ? f Xi ? f Xj? ?That is, the kernel function, represents a dot product of input data points mapped into the higher dimensional featurespace by transformationf. Gamma is an adjustable parameter of certain kernel functions. The RBF is by far the most popularchoice of kernel types used in Support Vector Machines. This is mainly because of their localized and finite responses acrossthe entire range of the real x-axis.3.5. Radial basis function (RBF)It is a real-valued function whose value depends only on the distance from the origin, so that f(X) = f(X); or alternativelyon the distance from some other point c, called a center, so that f(X,C) = f(X ? C). Any function f that satisfies the propertyf(X) = f(X) is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible.For example, using LukaszykKarmowski metric, it is possible for some radial functions to avoid problems with illconditioning of the matrix solved to determine coefficients wi, since the X is always greater than zero 16,17.Sums of radial basis functions are typically used to approximately estimate the given function. This approximationprocess can also be interpreted as a simple kind of network. Preetham et al., 2014 presented the state of art of support vectormechanics method (SVM) problems related to civil engineering. Areas of many research are ongoing numericalinvestigations on SVM techniques are shown. Many researches from RBFs are also used as a kernel in support vectorclassification 16.10 A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 8154. Results and discussion4.1. Propertiesof light weight foamed concreteLight weight foamed concrete samples in this work were tested for their fresh, dry density, 7 days and 28 dayscompressive strength. The parameters that are taken into account in mix proportions are those believed to affectcompressive strength of foamed concrete according to previous work and other researchers 710. It is believed that densityis the key factor affecting its compressive strength of foamed concrete as the amount of foam added to the mix controlling itsdensity and hence, its strength. to the mix. This is attributed to the fact that increasing the air bubbles induced by foamingagent added to the mix will increase porosity while weakening its strength. Fig. 1 illustrates this fact into which therelationship between compressive strength of light weight foamed concrete and its dry density.On the other hand, increasing cement content of light weight foamed concrete increases its compressive strength(especially because foamed concrete does not include coarse aggregates only fine aggregates) as shown in Fig. 2. The reasonexplaining this trend is, by increasing cement, the fine materials that react with water increases leading to more hydrationproducts and binding for the mix which increases the strength. Also, it was found that increasing particle size of fineaggregate decreases its strength (Table 1).4.2. First: traditional multivariable nonlinear regressionTo predict the 28 day compressive strength of foamed concrete, nonlinear regression was used to analyze a data set of 150samples. In this study, the main variables that were used to model the 28 days compressive strength were; density, cementcontent, sand content, w/c ratio, sand particle size, foaming agent, foam content, and the compressive strength at 7 days. Thegeneral model for the nonlinear regression for the compressive strength at 28 day was:Var10 = a0*v1a1*v2a2*v3a3*v4a4*v5a5*v6a6*v7a7*v8a8*v9a9Where:Var10 (Dependent variable) = compressive strength at 28 dayV1 to V9 (Independent variables) = the input parametersThe Loss function used in this analysis was least squares.When the actual observation compared with the predicted results using the developed model, the correlation coefficientwas found R = 0.97884248 and the r2 = 0.9581326The Level of confidence limit was: 95% (alpha = 0.050). The coefficient of model parameters (an) with the standarddeviation, t-value and p-value for each one were listed in Table 2.The actual observations plotted with the predicted results generated from the regression model in Fig. 3(a). This plotexplain the high correlation between the two data set and reflect the high accuracy of the developed model. There are fewpoints around the compressive strength of 30 MPa was little diverge from the actual observations. This may belong to theproperties of raw material (especially the particle size of sand) and the specific condition of the samples tested.Plotting the residual values with the predicted results reflect the good performance of the developed model. Again thisplot shows exactly the amount of errors for each reading in term of predicted compressive strength. The interval around the30 MPa is very clear here with some convergence from the actual observations, meanwhile the most results was of errorranging between (?4 and +4) as shown in Fig. 3(b). The overall correlation coefficient for the 9-inputs and the target outputshows that the highest correlations with the 28th day compressive strength was with the 7th day compressive strength,010203040501300 140015 00 160 0 170018 00 190 0 2000 2100Compressive Strength (MPa)Density (Kg/m3)Fig. 1. Compressive strength vs. density of foamed concrete.A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 815 11density, and cement content respectively in positive direction. It was correlated in negative direction with w/c ratio, sand/cement ration and foam content respectively.4.3. Second: support vector machineTo implement this technique, the compressive strength at 28 days was considered as the dependent variable(Var10), andthe other inputs (V1 to V9) as the independent variables. The sample size of 150 overall observations was randomly dividedinto Train of (111 samples), and Test of (39 samples). The support vector machine of type 1 was adopted for the analysisprocess. The four kernel function types were tested: Radial Basis Function, linear, polynomial, and sigmoid. This processresults were listed in Table 3.It was clear that the RBF has the best results in term of highest correlation for training, testing and the overall data set. Ithas the minimum mean square error amongst the four functions, and has the minimal standard deviation. So the detaileddiscussion will focus on the RBF to explain the main features of this model. Table 4 illustrates that the total error mean in thepredicted model was found around (?0.32084) for all investigated samples. The overall correlation coefficient was verysignificant (around 99%), which reflects the high degree of precision for the developed model as shown in Fig. 4. The0102030405060400 500 60070 0 800 900 1000Compressive Strength (MPa)Cement Content (Kg/m3)Fig. 2. Compressive strength vs. cement content of foamed concrete.Table 1Mix proportions details.Mix No. of Samples S/C ratio w/c ratio Density Range (Kg/m3) SP (L) Sand max Size (mm) Sand TypeC1 18 1 0.45 14002000 0 0.6 silica sandC2 18 0.5 0.45 14002000 0 0.6 silica sandC3 18 2 0.45 14002000 0 0.6 silica sandC4 18 1 0.4 14002000 0 0.6 silica sandC5 18 1 0.35 14002000 0.2 0.6 silica sandC6 18 1 0.3 14002000 0.3 0.6 silica sandC7 18 1 0.35 14002000 0.3 0.6 river wash sandC8 18 1 0.35 14002000 0.4 1.18 river wash sandC9 6 1 0.4 14002000 0.5 4.75 river wash sandTable 2The coefficient of regression model (parameters a).Parameter variable Estimate Standard ? error t-value ? df = 140 p-valuea0 0.000041 0 0.33857 0.735442a1 density 1.13292 0.58 1.9684 0.050997a2 cement 0.33708 61506.59 0.00001 0.999996a3 sand 0.338127 61506.55 0.00001 0.999996a4 sand/cement ?0.1796 61506.57 0 0.999998a5 water/cement 0.328483 0.15 2.24687 0.026213a6 sand-size ?0.21411 0.03 ?7.50364 0a7 Agent ?0.07656 0.02 ?3.81834 0.000201a8 foam 0.107572 0.06 1.72006 0.087631a9 comp-7D 0.5375 0.05 11.1801 012 A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 815(a) Compressive strength (Predicted vs. Observed)(b) Compressive strength (Predicted results vs. Residual)05101520253035404550550 5 10 15 20 25 30 35 40 45 50 55Observed Value (MPa)Predi cted Valu e (MPa)-10-8-6-4-202468101214160 10 20 30 40 50Residual Value (MPa)Predicted Value (MPa)Fig. 3. Properties of regression model.Table 3Support Vector machine the four type functions results.Function Type Correlation coefficient Mean square error Standard deviationRadial Basis Function 0.986(Train), 0.990(Test), 0.987(Overall 3.880(Train), 3.268(Test), 3.721(Overall) 0.170(Train), 0.147(Test), 0.165(Overall)linear 0.951(Train), 0.945(Test), 0.949(Overall) 18.444(Train), 25.263(Test), 20.217(Overall) 0.369(Train), 0.413(Test), 0.381(Overall)Polynomial 0.976(Train), 0.986(Test), 0.978(Overall) 6.714(Train), 5.357(Test), 6.361(Overall) 0.225(Train), 0.178(Test), 0.215(Overall)Sigmoid 0.851(Train), 0.877(Test), 0.859(Overall) 67.969(Train), 66.761(Test), 67.655(Overall) 0.716(Train), 0.673(Test), 0.703(Overall)Table 4Main features of the RBF support vector machine model.Number of support vectors 30 (16 bounded), (gamma = 0.111)model specifications (decision constants) 0.124238Observed mean 26.90346Predictions mean 27.22430Observed S.D. 12.28756Predictions S.D. 11.54473Mean squared error 3.26776Error mean ?0.32084Error S.D. 1.80225Abs. error mean 1.49713S.D. ratio 0.14667Correlation 0.99A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 815 13observed data set was plotted against predicted values of the RBF for the train data set (a), test data set (b) and overall data set(c). It is very clear that the predicted values distributed very close to the equality line for all sets of data in figures and highlycorrelated to the actual observed data which indicates strong reliability of the proposed above model. For each data set theformula for the best fitting was provided with its plot.5. ConclusionsThe results revealed from this work includes the effect of mix proportions on 28 day compressive strength of light weightfoamed concrete. The positive effect of density and cement content was very clear and gave evidence that these two factorshave important and significant role in designing foamed concrete mixes. Meanwhile, higher negative impact on thecompressive strength of foamed concrete was proved to be with increasing in w/c ratio, sand/cement ratio and foam contentrespectively.A mathematical model for the prediction of the lightweight foamed concrete compressive strength was proposed in thisstudy. The technique used to perform the proposed model was traditional multivariable nonlinear regression and(a) Train data sety = 0.3771+0.9765x05101520253035404550550 5 10 15 20 25 30 35 40 45 50 55Predicted 28-day Compressive strength (MPa)Observed 28-day Compressive strength (MPa)(b) Test data set(c) Overall data sety = 0.122 6+0.994 4x 05101520253035404550550 5 10 15 20 25 30 35 40 45 50 55Predicted 28-day Compressive strength (MPa)Observed 28-day Compressive strength (MP a)y = 1.6018+0.9419x05101520253035404550550 5 10 15 20 25 30 35 40 45 50 55Predicted 28-day Compressive strength (MPa)Observed 28-day Compressive strength (MP a)Fig. 4. Correlation plots for (a) train data, (b) test data, and (c) Overall data set.14 A.M. Abd, S.M. Abd / Case Studies in Construction Materials 6 (2017) 815revolutionary Support Vector Machine modelling. The results revealed excellent correlation between the observed andpredicted values for the data set used in this study. Both techniques proved to be attractive tool for the prediction process.The S